INVARIANT TWINS
necessary, although there is a simpler way of reasoning the
result out, without actually calculating A, B, C) that
J52 - AC = (ps - gr)2 (&" - ac).
Now b2 — ac is called the discriminant of the quadratic equation
in x\ hence the discriminant of the quadratic in y is B3 — AC>
and it has been shown that the discriminant of the transformed
equation is equal to the discriminant of the original equation, times
(he factor (ps — gr)2 which depends only upon the coefficients
p, g, r, s in the transformation y = (px + q)l(rx + s) by means
of which x was expressed in terms ofy.
Boole was the first (in 1841) to observe something worth
looking at in this particular trifle. Every algebraic equation has
a discriminant, that is, a certain expression (such as &3 — ac for
the quadratic) which is equal to zero if, and only if, two or more
roots of the equation are equal. Boole first asked, does the dis-
criminant of every equation when its x is replaced by the related
y (as was done for the quadratic) come back unchanged except
for a factor depending only on the coefficients of the transfor-
mation? He found that this was true. Next he asked whether
there might not be expressions other 'than discriminants con-
structed from the coefficients having this same property of
invariance under transformation. He found two such for the
general equation of the fourth degree. Then another man, the
brilliant young German mathematician, F. M. G. Eisenstein
(1823-52) following up a result of Boole's, in 1844, discovered
that certain expressions involving both the coefficients and the x
of the original equations exhibit the same sort of invariance: the
original coefficients and the original x pass into the transformed
coefficients and y (as for the quadratic), and the expressions in
question constructed from the originals differ from those con-
structed from the transforms only by a factor which depends
solely on the coefficients of the transformation.
Neither Boole nor Eisenstein had any general method for
finding such invariant expressions. At this point Cayley entered
the field in 1845 with his pathbreaking memoir, On the Theory of
Linear Transformations. At the time he was twenty-four. He
set himself the problem of firming uniform methods which
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